# Lorentz Oscillator and dielectric function: Incorrect formulation in textbook?

According to what I was taught in school the relative dielectric function for a Lorentz-Oscillator can be formulated as:

$$\epsilon(\omega)=\epsilon_\infty+\frac{\omega_p^2}{\omega_0^2-\omega^2+j\gamma\omega }$$

where $$\epsilon_\infty$$ is a constant and corresponds to the high-frequency limit for the dielectric function, $$\omega_p$$ is the plasma frequency, $$\omega_0$$ is the resonance frequency and $$\gamma$$ is a damping parameter. The derivation of this formula is straight forward and can be found, for example in the MIT lecture notes here on page 8.

Now, the authors of the most popular textbook on computational electrodynamic, called "computational electrodynamic - The finite difference Time-domain method" by Allen Taflove and Susan Hagness (3rd ed) claim that the relative permittivity for a Lorentz-medium is

$$\epsilon(\omega)=\epsilon_\infty+\frac{\Delta \epsilon \omega_p^2}{\omega_p^2+2j\omega\gamma-\omega^2}$$

Even by using the relation $$\Delta\epsilon=\frac{\omega_p^2}{\omega_0^2}$$, the closest I can come, starting from the 1st equation is

$$\epsilon(\omega)=\epsilon_\infty+\frac{\Delta \epsilon \omega_p^2}{\omega_p^2+\Delta\epsilon j \gamma\omega-\Delta\epsilon\omega^2}$$

I don't see how an algebraic manipulation allows me to arrive at the equation provided by Taflove. Is there a physical detail I am missing or is Taflove simply wrong or is there another explanation?

EDIT: I just recognized that the $$\gamma$$ as used by Taflove is not the same as the usual $$\gamma$$. However, he just calls it "damping parameter" without any derivation. So, by assuming the equations are equivalent these parameters are related by $$\gamma_\text{Taflove}=\frac{1}{2}(\Delta\epsilon\gamma+\Delta\epsilon j \omega-j\omega)$$.

Obviously $$\gamma_\text{Taflove}$$ can become imaginary...So I'm wondering why the hell would somebody come up with such a formulation, where the alternative "damping parameter" loses any physical meaning in contrast to the original one, which might be interpreted as "number of collision" of some particles/quasiparticles having units of 1/s. I don't see any benefit from the second formulation at all. Am I missing something?

• Are the definitions of $\gamma$ the same in both cases? – garyp Aug 8 at 13:10
• Good hint. I'm checking it... – OD IUM Aug 8 at 13:17
• The Taflove expression doesn't seem to resonate near $\omega_0$, but I'm only using my eye. I don't have paper and pencil nearby. – garyp Aug 8 at 15:24